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1. International
OPEN
Journal
ACCESS
Of Modern Engineering Research (IJMER)
On Contra – RW Continuous Functions in Topological Spaces
M. Karpagadevi1, A. Pushpalatha2
1
Assistant Professor, S.V.S. College of Engineering, Coimbatore, India
2
Assistant Professor, Government Arts College, Udumalpet, India
ABSTRACT: In this paper we introduce and investigate some classes of functions called
contra-rw continuous functions. We get several characterizations and some of their properties. Also we
investigate its relationship with other types of functions.
Mathematics Subject Classification: 54C08, 54C10
Keywords: Contra rw-continuous, Almost Contra rw-continuous, Contra rw-irresolute
I. Introduction
In 1996, Dontchev[1] presented a new notion of continuous function called contra-continuity. This
notion is a stronger form of LC-Continuity. In 2007, Benchalli.S.S. and Wali.R.S.[2] introduced RW-closed set
in topological spaces. In 2012, Karpagadevi.M. and Pushpalatha.A.[3] introduced RW-continuous maps and
RW-irresolute maps in topological spaces. The purpose this paper is to define a new class of continuous
functions called Contra-rw continuous functions and almost Contra rw-Continuous function and investigate their
relationships to other functions.
II. Preliminaries
Throughout this paper X and Y denote the topological spaces (X,) and (Y,σ) respectively and on
which no separation axioms are assumed unless otherwise explicitly stated. For any subset A of a space (X,),
the closure of A, the interior of A and the complement of A are denoted by cl(A), int(A) and Ac respectively.
(X,) will be replaced by X if there is no chance of confusion. Let us recall the following definitions as pre
requesters.
Definition 2.1: A Subset A of (X,) is called
(i) generalized closed set (briefly g-closed) [5] if cl(A)  U whenever A  U and U is open in X.
(ii) regular generalized closed set (briefly rg-closed)[6] if cl(A)  U whenever A  U and U is regular open in
X.
The Complement of the above mentioned closed sets are their respective open sets.
Definition 2.2: A Subset A of a Space X is called rw-closed [2] if cl(A)  U whenever A  U and U is regular
semiopen in X.
Definiton 2.3: A Map f: (X,) → (Y,σ) is said to be
(i) rw-continuous[3] if f-1(V) is rw-closed in (X,) for every closed set V in (Y,σ)
(ii) rw-irresolute[3] if f-1(V) is rw-closed in (X,) for every rw-closed set V of (Y,σ)
(iii) rw-closed[2] if f(F) is rw-closed in (Y,σ) for every rw-closed set F of (X,)
(iv) rw-open[2] if f(F) is rw-open in (Y,σ) for every rw-open set F of (X,)
Definiton 2.4: A Map f: (X,) → (Y,σ) is said to be contra-continuous [1] if f-1 (V) is closed in (X,) for
every open set V in (Y,σ).
III. Contra RW-Continuous Function
In this section we introduce the notions of contra rw-continuous, contra rw-irresolute and almost contra
rw-continuous functions in topological spaces and study some of their properties.
Definition 3.1: A function f: (X,) → (Y,σ) is called contra rw-continuous if f-1 (V) is rw-closed set in X for
each open set V in Y.
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2. On Contra – RW Continuous Functions in Topological Spaces
Example 3.2: Let X=Y={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ = {Y,,{a,b}}. Let f: X → Y
be a map defined by f (a) = a, f (b) = b and f(c) = c. Clearly f is contra rw-continuous.
Theorem 3.3: Every contra-continuous function is contra rw-continuous.
Proof: The proof follows from the fact that every closed set is rw-closed set.
Remark 3.4: The converse of the above theorem need not be true as seen from the following example.
Example 3.5: Let X=Y={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ = {Y,,{a,b}}. Let f: X → Y
be a map defined by f (a) = a, f (b) = b and f(c) = c. Clearly f is contra rw-continuous but not contra-continuous
since f -1({a,b}) = {a,b} which is not closed in X.
Theorem 3.6: If a function f: X → Y is contra rw-continuous, then f is contra-continuous.
Proof: Let V be an open set in Y. Since f is contra rw-continuous, f -1(V) is closed in X. Hence f is contracontinuous.
Remark 3.7: The concept of rw-continuity and contra rw-continuity is independent as shown in the following
examples.
Example 3.8 : Let X=Y={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ = {Y,,{b,c}}. Let f: X → Y
be a map defined by f (a) = a, f (b) = b and f(c) = c. Clearly f is contra rw-continuous but not rw-continuous
since f -1({a}) = {a} is not rw-closed in X where {a} is closed in Y.
Example 3.9 : Let X=Y={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ = {Y,,{b}}. Define f: X → Y
identity mapping. Then c learly f is rw-continuous but not contra rw-continuous since f -1({b}) = {b} is not
rw-closed in X where {b} is closed in Y.
Theorem 3.10: Every contra rw-continuous function is contra rg-continuous.
Proof: Since every rw-closed set is rg-closed, the proof is obvious.
Remark 3.11: The converse of the above theorem need not be true as seen from the following example.
Example
3.12:
Let
X=Y={a,b,c,d}
with
topologies
={X,,{a},{b},{a,b},{a,b,c}}
and
σ = {Y,,{a},{b},{a,b,c}}. Let f: X → Y be a map defined by f (a) = c, f (b) = d, f(c) = a and f (d) =b. Clearly f
is contra rg-continuous but not contra rw-continuous since f -1({a}) = {c} is not rw-closed in X where {a} is
open in Y.
Remark 3.13: The composition of two contra rw-continuous functions need not be contra rw-continuous as
seen from the following example.
Example 3.14: Let X=Y=Z={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ = {Y,,{a,b}} and
 = {Z, ,{a}}. Let f: X → Y be a map defined by f (a) = c, f (b) = b and f(c) = a and g: Y→Z is defined by
g (a) =b, g (b) =c and g(c) =a.Then clearly f and g are contra rw-continuous. But g  f: X → Z is not contra
rw-continuous since (g  f)-1{a} = f-1(g-1{a}) = f-1 ({c}) = {a} is not rw-closed in X.
Theorem 3.15: If f: (X,) → (Y,σ) is contra rw-continuous and g: Y→Z is a continuous function, then
g  f : X→Z is contra rw-continuous.
Proof: Let V be open in Z. Since g is continuous, g -1(V) is open in Y. Then f -1(g -1(V)) is rw-closed in X since f
is contra rw-continuous. Thus (g  f) -1 (V) is rw-closed in X. Hence g  f is contra rw-continuous.
Theorem 3.16: If f: X → Y is rw-irresolute and g: Y→Z is contra continuous function then g  f: X→Z
contra rw-continuous.
Proof: Since every contra continuous is contra rw-continuous, the proof is obvious.
is
Theorem 3.17: If f: X → Y is contra rw-continuous then for every x X, each F C(Y,f(x)), there exists
URWO(X,x) such that f(U)  F (ie) For each x X, each closed subset F of Y with f(x)  F, there exists a
rw-open set U of Y such that xU and f(U)F.
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3. On Contra – RW Continuous Functions in Topological Spaces
Proof: Let f: X → Y be contra rw-continuous. Let F be any closed set of Y and f(x)  F where xX. Then
f -1(F) is rw-open in X. Also x  f -1(F). Take U = f -1(F). Then U is a rw-open set containing x and f (U)  F.
Theorem 3.18: Let (X,) be a rw-connected space (Y,σ) be any topological space. If X → Y is surjective and
contra rw-continuous then Y is not a discrete space.
Proof: Suppose Y is discrete space. Let A be any proper non-empty subset of Y. Then A is both open and
closed in Y. Since f is contra rw-continuous, f -1(A) is both rw-open and rw-closed in X. Since X is
rw-connected, the only subset of Y which are both rw-open and rw-closed are X and. Hence f -1(A) = X. Then
it contradicts to the fact that f: X → Y surjective. Hence Y is not a discrete space.
Definition 3.19: A function f: X → Y is called almost contra rw-continuous if f -1(V) is rw-closed set in X for
every regular open set V in Y.
Example 3.20: Let X=Y=Z={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ ={ Y,,{a},{c},{a,c}}.
Define f: X → Y by f (a) =a, f (b) =b and f(c)=c. Here f is almost contra rw-continuous since f -1({a}) = {a} is
rw-closed in X for every regular open set {a} in Y.
Theorem 3.21: Every contra rw-continuous function is almost contra rw-continuous.
Proof: The proof follows from the definition and fact that every regular open set is open.
Definition 3.22: A function f: X → Y is called contra rw-irresolute if f -1(V) is rw-closed in X for each rw-open
set V in Y.
Definition 3.23: A function f: X → Y is called perfectly contra rw-irresolute if f -1(V) is rw-closed and rw-open
in X for each rw-open set V in Y.
Theorem 3.24: A function f: X → Y is perfectly contra rw-irresolute if and only if f is contra rw-irresolute and
rw-irresolute.
Proof: The proof directly follows from the definitions.
Remark 3.25: The following examples shows that the concepts of rw-irresolute and contra rw-irresolute are
independent of each other.
Example 3.26: Let X=Y=Z={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ ={ Y,,{a},{c},{a,c}}.
Define f: X → Y by f (a) = a, f (b) = b and f(c) = c. Here f is contra rw-irresolute but not rw-irresolute since
f -1({a,b}) = {a,b} is not rw-open in X.
Example 3.27: Let X=Y=Z={a,b,c} with topologies  = {X,,{a},{b},{a,b}} and σ ={ Y,,{a},{c},{a,c}}.
Define f: X → Y by f (a) = a, f (b) = b and f(c) = c. Here f is rw-irresolute but not contra rw-irresolute since
f -1({b}) = {b} is not rw-closed in X.
Theorem 3.28: Let f: X → Y and g: Y → Z be a function then
(i) If g is rw-irresolute and f is contra rw-irresolute then g  f is contra rw-irresolute
(ii) If g is contra rw-irresolute and f is rw-irresolute then g  f is contra rw-irresolute
Proof: (i) Let U be a rw-open in Z. Since g is rw-irresolute, g -1(U) is rw-open in Y. Thus f -1(g -1(U)) is
rw-closed in X. Since f is contra rw-irresolute (ie) (g  f) -1(U) is rw-closed in X. This implies that g  f is contra
rw-irreolute.
(ii) Let U be a rw-open in Z. Since g is contra rw-irresolute, g -1(U) is rw-closed in Y. Thus f -1(g -1(U)) is
rw-closed in X since f is rw-irresolute (ie) (g  f) -1(U) is rw-closed in X. This implies that g  f is
contra rw-irresolute.
Theorem 3.29: Every perfectly contra rw-irresolute function is contra rw-irresolute and rw-irresolute.
Proof: The proof directly follows from the definitions.
Remark 3.30: The following two examples shows that a contra rw-irresolute function may not be perfectly
contra rw-irresolute and rw-irresolute function may not be perfectly contra rw-irresolute.
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4. On Contra – RW Continuous Functions in Topological Spaces
Example 3.31: In example 3.29, f is rw-irresolute but not perfectly contra rw-irresolute and in example 3.28,
f is contra rw-irresolute but not perfectly contra rw-irresolute.
Theorem 3.32: A function is perfectly contra rw-irresolute iff f is contra rw-irresolute and rw-irresolute.
Proof: The proof directly follows from the definitions.
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| IJMER | ISSN: 2249–6645 |
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